If your sub-site is inheriting navigation from Parent site, the link "Edit Links" in top navigation will not appear. To enable edit links SharePoint 2013 - Your site shouldn't inherit navigation from the parent site.

Its available only for "Structural Navigation" option under "Navigation Settings" on publishing sites.

You may have custom CSS deployed which hides "Edit Links"!

How to remove edit links in SharePoint 2013?
Use this CSS in either at Master page or in Script/Content editor web parts to hide edit links in SharePoint 2013.

4 comments
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In case the site an incorrect site template was used, JavaScript is probably the only way round to sorting the menu, here is an example..http://tusharwithsharepoint.blogspot.co.uk/2016/03/top-navigation-wont-save.html